Various materials are known to exhibit the Villari effect, that is, their permeability μ varies with the stress (force) applied. The Villari effect can be thought of as the inverse of the magnetostrictive effect, which is a change in dimension of a material under applied magnetic field. These materials have been used in various configurations to make force sensors. A simple configuration for a force sensor 10, that will serve here as exemplary, consists of a coil 12 wound around a shaft 14 made of the magnetostrictive material, as shown in FIG. 1. The magnitude of the voltage Vcoil of the coil 12 is related to its inductance L as follows:|Vcoil|=2π f I L(μ)=2π f I L(F)  (1)where f is the frequency in Hertz and I the magnitude of the sinusoidal current impressed on the coil 12, and L is the coil inductance. Inductance L, as shown in Eq. 1, is a function of permeability μ or force F.
The circuit shown in FIG. 1A can be used for the measurement of dynamic inductance, for instance as part of a magnetostrictive force sensing apparatus. Referring to the circuit in FIG. 1A, L is the varying inductance, and R and RL are fixed resistances. The voltage V is imposed at a frequency f chosen for best sensitivity. The measurement can be performed by measuring the change across a portion of the voltage divider (R) as Vout. The inductance L is related to the magnitudes of the applied and measured voltages, |V| and |Vout|, respectively, by the following formula:   L  =            R              2        ⁢        π        ⁢                                  ⁢        f              ⁢                            1                      k            2                          -                              [                                                            R                  L                                R                            +              1                        ]                    2                    where:   k  =                          V        out                                V            The output voltage Vout can be processed by a microprocessor or other similar circuit to obtain a signal that is representative of the change in inductance L. Any non-linearity in the inductance-versus-force function can be also included in the algorithm to provide the desired force measurement output. Compensation factors for temperature variations, etc, may also be included. The resistance values, R and RL, may advantageously be chosen according to the average value of inductance L, frequency f and source V.
The inductance L for a given number of coil turns N is generally a function of the permeability μ of the shaft 14, and the length l and cross-section A of the magnetic flux path around coil 12:                     L        =                              μ            ⁢                                                  ⁢                          N              2                        ⁢            A                    l                                    (        2        )            
The magnetic field created by the coil 12 thus includes the shaft 14, but also includes a return path as shown schematically by the magnetic flux lines 16 in FIG. 2. Eq. 2 must therefore be expressed as the sum of two terms, one for the shaft 14, indicated by subscript “sh”, and one for the return path 16, indicated by subscript “ret”:                     L        =                              N            2                    ⁡                      [                                                                                μ                    sh                                    ⁢                                      A                    sh                                                                    l                  sh                                            +                                                                    μ                    ret                                    ⁢                                      A                    ret                                                                    l                  ret                                                      ]                                              (        3        )            
From Eq. 3, it is clear that the sensor signal is a function of the surrounding material, that is, of the environment around the sensor 10. The signal will differ when surrounded by a magnetic material versus a non-magnetic material or air. Therefore, sensors must currently be designed to take into account the environment in which the sensor is to be used. It would be desirable to devise a sensor in such a way as to remove the dependency on the properties of the surrounding material.
In addition, to obtain a large signal, a material with large magnetostrictive behavior must be chosen for the shaft 14, i.e., a large μ variation (μmax−μmin) for a given stress (force) change. However, the average value of μsh can also affect the signal. Also, since many magnetostrictive materials are conductive, eddy currents will be induced in the shaft 14, which will restrict the magnetic field towards the outer surface 14a of the shaft 14, thus reducing the effective cross section Ash of the shaft 14. In that sense, a high frequency f results in a larger signal (f is a multiplying factor in Eq. 1), but generates more eddy currents and further restricts the magnetic field to the shaft surface 14a. Thus, design trade-offs are necessary. Therefore, it is further desirable to provide preferred combined values of operating frequency f average permeability μsh and resistivity ρ for the magnetostrictive material.